Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Computable many-one reductions between languages
Definition
Let and be languages over finite alphabets. A computable many-one reduction from to is a total computable function such that for every one has
When such an exists, write
Remarks
-
The reduction sends each source instance to one target instance.
-
The biconditional preserves both yes-instances and no-instances.
Depends on
Used by
- A reduction to the complement is not a reduction to the original target Counterexample
- Computably enumerable hardness and completeness under computable many-one reductions Definition
- FALSE: a valid many-one reduction may reverse yes-instances and no-instances False statement
- Computable many-one reducibility is reflexive and transitive Lemma
- Every computable many-one reduction induces a Turing reduction Lemma
- Turing reducibility is strictly weaker than computable many-one reducibility Proposition
- CFG ambiguity is undecidable Theorem
- CFG universality is undecidable Theorem
- Computable many-one reductions transfer decidability and recognizability backward Theorem
- Emptiness of the intersection of two CFGs is undecidable Theorem
- Every nontrivial extensional property of Turing-machine languages is undecidable Theorem
- The modified Post correspondence problem is undecidable Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MIT 6.045J / 18.400J, Lecture 9: Mapping Reducibility and Rice's Theorem (standard reference, not scraped)
- Kevin Kelly, Many-one Reduction (standard reference, not scraped)